This pocket-size book introduces set theory and its applications to computing, data, and engineering. We start with basic concepts such as membership, equality, subsets, relations, and functions. We then consider equivalence, order, cardinality, induction, and recursion. We also discuss infinite sets and the axioms of set theory, along with their consequences for representation and computation.
The applications include Boolean logic, relational databases, programming collections and types, formal specification with Z, B, and Alloy, combinatorial counting, probability, and graphs. The same mathematical concepts apply across contexts. For example, the distinction between sets and multisets helps explain how SQL handles duplicates. Relations model dependencies and connectivity. Different independence assumptions lead to different conclusions about collision probabilities. We also consider fuzzy sets, rough sets, and multisets as generalizations that require separate examination of their operations and laws.
We use definitions, theorems, and worked examples throughout. We pay particular attention to the assumptions behind mathematical results and the differences between abstract models and their software implementations.
The book is intended for software developers, data analysts, engineers, and students interested in the mathematical foundations of their work. A notation reference, an annotated bibliography, and an index are included for further study and practical reference.
